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Acta Physica Polonica A

Probing Higher Dimensions of Hilbert Space in Experiment

Anton Zeilinger

Published Apr 1, 1994• 4 citations
Publisher

Abstract

Dedicated to Prof. Iwo Białynicki-Birula on the occasion of his 60th birthdayUsing the correlated pairs produced in spontaneous parametric down--conversion, one can extract quantum states effectively defined in a Hilbert space of any dimension N. Furthermore, using just beam splitters and phase shifters one can build any unitary operator in the laboratory.We briefly discuss how this can be done, what kind of states could easily be produced in the laboratory, and we will discuss one explicit result pertaining to photon bunching in an N-dimensional Hilbert space.

DOI
10.12693/aphyspola.85.717

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Acta Physica Polonica A•Apr 1, 1994•1 citation
Dedicated to Prof. Iwo Białynicki-Birula on the occasion of his 60th birthdayUsing the correlated pairs produced in spontaneous parametric down--conversion, one can extract quantum states effectively defined in a Hilbert space of any dimension N. Furthermore, using just beam splitters and phase shifters one can build any unitary operator in the laboratory.We briefly discuss how this can be done, what kind of states could easily be produced in the laboratory, and we will discuss one explicit result pertaining to photon bunching in an N-dimensional Hilbert space.
Enhancement of entanglement enhancement is necessary for most quantum communication protocols many of which are defined in Hilbert spaces larger than two. In this work we present the experimental realization of entanglement concentration of orbital angular momentum entangled photons produced in the spontaneous parametric down-conversion process which have been shown to provide a source for higher dimensional entanglement. We investigate the specific case of three dimensions and the possibility of generating different entangled states out of an initial state. The results presented here are of importance for pure states as well as for mixed states.
Multi-particle entangled states are becoming more and more interesting for different applications in quantum information, such as quantum multi-particle teleportation, entangled based cryptography and quantum communication schemes based on multi-particle states, but also for fundamental reasons such as testing local realism by means of the Greenberger-Horne-Zeilinger states (GHZ) arguments . The advantage of multiparticle entanglement versus the most common two particle one lies in the fact that the greater the number of particles involved in the entangled state, the more clearly the quantum effects are exhibited. Our final goal is to exploit the multi-photon entangled states for quantum metrology. By the interaction of those states with nanostructures (mostly nanoholes, but in a future also nanospheres), we expect to reach sub-wavelength sensitivity phase measurements beating the Standard Quantum Limit as well as to detect plasmonic effects. In this thesis we present our work consisting on designing and building an experimental setup for the generation of states of 4-photon entangled in the transverse linear momentum. In the first part we explain some basics in the form of a technical introduction. We then adapt a general theoretical model to our particular case, explaining the physics of the process of generation of entangled states of photons. We proceed explaining the details of our setup, as well as the procedure of measurement. We then present the results of our tests in a "standard" 2-photon regime to show the correct functioning of the setup itself. Some more sophisticated measurements using structured light in the form of a first-order Hermite-Gaussian mode has been performed to generate entanglement in higher spatial modes are also presented. We conclude with theoretical and experimental details regarding the 4-photon experiments feasible with our setup. Our source is based on Spontaneous Parametric Down-Conversion (SPDC). In the last two decades it has been extensively studied for the generation of two-photon entangled states, since Kwiat et al. published in 1995 the new scheme based on polarization entangled photons. Most of the studies regarding entangled states involving more than two photons are indeed based on quantum correlations in the polarization or in the path of the down converted beams. The great advantage of entangling a spatial degree of freedom (as we do) is related to the dimensionality of the Hilbert space in which the state is defined. In the particular case of linear momentum, which is a continuous variable, we are dealing with infinite dimensions. Since in each dimension of the state one quanta of information can be encoded, the advantage of having multidimension entanglement for the field of quantum communication and computing is clear. The specs of our setup include a periodically-poled KTP crystals offering high conversion efficiency and cut for type-0 SPDC. The photons generated are well known to be inherently entangled in different degrees of freedom such as polarization, energy-time, linear momentum, and orbital angular momentum. Since each single SPDC event generates one photon pair, in order to reach four photon we entangle two consecutive SPDC events per each state. Furthermore, it is possible to manipulate (to enhance) the dimensionality of the entanglement by properly tailoring the pump spatial profile. The key point for the generation of four-photon entangled states is that, under some conditions, two consecutive pairs generated via SPDC cannot be described as two independent pairs but necessarily as a four-photon state showing genuine entanglement. This can happen when using a pulsed laser to generate the entangled photons; in particular, in the regime in which the coherence time of the down-converted photons τc is (much) longer than the temporal length of the pumping pulse Δt. In this condition the emission of a second pair is stimulated by the presence of the first one. That effect can be ultimately seen as a constructive multi-particle interference effect, the deep origin of which has to be sought in the boson statistic of photons.
In this thesis we explore spatial quantum correlations of high-dimensional multi-photon states. These states are produced using the process of parametric down-conversion and are experimentally explored by measuring correlations with only two detectors. Compared to earlier investigations of multi-photon states, the correlations in this thesis are created in the spatial domain instead of the temporal domain. This has a distinct experimental advantage because it is much easier to measure the emission direction compared to a measurement of the arrival time of the photons.
Dedicated to Prof. Iwo Białynicki-Birula on the occasion of his 60th birthdayThe interaction of multi-level atoms with several to many lasers simultaneously is examined in two regimes, the "short" regime of resonant atomic response to prescribed laser fields, and the "long" regime of resonant pulse propagation in which several to many laser pulses pass through an active medium composed of many-level atoms.Results obtained with I. Białynicki-Birula et al. in the short regime are reviewed, and new results in the long regime are described.
Dedicated to Prof. Iwo Białynicki-Birula on the occasion of his 601h birthdayIn earlier work universal propagators were introduced for the Heisenberg -Weyl group, the affine group, and the rotation group.By generalizing these constructions we show here that it is possible to introduce a universal propagator for a rather general unitary Lie group.In the context of coherent-state representations, the universal propagator is a single function independent of any particular choice of fiducial vector, which nonetheless, propagates all coherent state Hilbert space representatives correctly.